# Math Help - arithmetic progression

1. ## arithmetic progression

Hiya, I was just having some trouble with this qn.

An AP has first term 75 and nth term -72. Sum of its first n terms is 33. Find n(i got 22-and i'm sure it's right-check if you might). Given that the sum from the (k+1)th term to the (2k)th term of the progression is positive, find k's greatest value.

(Ans:7 but how to get it?) Anyway thx a mil to whoever set up this forum...it's great! :P

2. Originally Posted by zeromeyzl
Hiya, I was just having some trouble with this qn.

An AP has first term 75 and nth term -72. Sum of its first n terms is 33. Find n(i got 22-and i'm sure it's right-check if you might). Given that the sum from the (k+1)th term to the (2k)th term of the progression is positive, find k's greatest value.

(Ans:7 but how to get it?) Anyway thx a mil to whoever set up this forum...it's great! :P
n = 22 is correct.

Then -72 = 75 + 21d => d = -2.

So the first term is 75 and the common difference is -2. Therefore $t_k = 75 + (k-1)(-2) = 77 - 2k$.

Therefore:

$S_{2k} - S_k = \frac{2k(75 + [77 - 2(2k)])}{2} - \frac{k(75 + [77 - 2k])}{2}$

$= \frac{2k(152 - 4k)}{2} - \frac{k(152 - 2k])}{2} = k(152 - 4k) - k(76 - k) = -3k^2 + 76k$.

So solve $-3k^2 + 76k > 0 \Rightarrow k(3k - 76) < 0$ for the largest integer value of k.

I get k = 25.